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Non-transitive Grime Dice, via Mathematica (latkin.org)
28 points by latkin on Jan 16, 2015 | hide | past | web | favorite | 5 comments

Here is what may surprise some people. Here is an expectation table (i.e. just averages):

    | :olive | :magenta | :blue | :yellow |  :red |
    |  4.167 |    4.333 |   4.5 |   4.667 | 4.833 |
Generated by this Clojure code (`exp` means expectation):

    (print-table (order-keys-by exp dice) [(exp dice)])
I'm surprised that the two (very good!) articles ([1] and [2]) I've read did not point that the non-transitive property [3] holds on the dice even though the expectation are transitive:

    E(olive) < E(magenta) < E(blue) < E(yellow) < E(red)
Of course the expectations have to be transitive; they are scalars.

When you apply a function to pairs (e.g. compare one die against another), you can get non-transitive behavior. This is not earth-shattering, but it is interesting.

Put another way: this is yet another reason to not trust a single summary statistic (e.g. the average in this case) when you really should look at the distribution.

My code is here: https://gist.github.com/xpe/30ae93b107c91ec2ccf5

(Edited at 12:57 PM EST.)

[1] OP: http://latkin.org/blog/2015/01/16/non-transitive-grime-dice-...

[2] http://www.singingbanana.com/dice/article.htm

[3] Actually, there are multiple cycles; the 'secondary' cycles are not as 'strong'.

Not non-intuitive at all. Reminds me of Rock-paper-scissors-lizard-Spock but in dice form.

Yes, the pairwise comparisons form a similar pattern as shown in the top-right pentagram in http://latkin.org/blog/wp-content/uploads/2015/01/cycles12.p... (each die beats two others and loses to two others, in pairwise comparisons)

The fact that rolling doubles causes some relationships to reverse (but some to stay the same) is very interesting and non-obvious, at least to me.

Good point. As an example:

    olive tends to win "blue vs olive" rolls.

    blue+blue tends to win "blue+blue vs olive+olive" rolls.
I uploaded blue vs olive histograms at http://imgur.com/a/p4zK8 -- note, for the comparison ones (labeled with "vs"), -1 means that the leftmost (first) roll won, 0 means a tie, and +1 means that the rightmost (last) roll won.

P.S. Blue vs Magenta here: http://imgur.com/a/9e6ne

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